N-free extensions of posets.Note on a theorem of P.A.Grillet

Details N-free extensions of posets.Note on a theorem of P.A.Grillet N-free extensions of posets.Note on a theorem of P.A.Grillet

2005-09-13

arxiv.org/abs/cs/0509034v1

Computer Science

Discrete Mathematics

7 pages, 4 pictures

Scientific paper

Let $S\_{N}(P)$ be the poset obtained by adding a dummy vertex on each diagonal edge of the $N$'s of a finite poset $P$. We show that $S\_{N}(S\_{N}(P))$ is $N$-free. It follows that this poset is the smallest $N$-free barycentric subdivision of the diagram of $P$, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with $P\_0:=P$ and consisting at step $m$ of adding a dummy vertex on a diagonal edge of some $N$ in $P\_m$, proving that the result of this algorithm does not depend upon the particular choice of the diagonal edge choosen at each step. These results are linked to drawing of posets.

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